The Sum and Difference of Two Constant Elasticity of Variance Stochastic Variables

Author(s)
Chi-Fai Lo

Affiliation(s)

Department of Physics, Institute of Theoretical Physics, The Chinese University of Hong Kong, Hong Kong, China.

Department of Physics, Institute of Theoretical Physics, The Chinese University of Hong Kong, Hong Kong, China.

ABSTRACT

We have applied the Lie-Trotter operator splitting
method to model the dynamics of both the sum and difference of two correlated
constant elasticity of variance (CEV) stochastic variables. Within the
Lie-Trotter splitting approximation, both the sum and difference are shown to
follow a shifted CEV stochastic process, and approximate probability distributions
are determined in closed form. Illustrative numerical examples are presented to
demonstrate the validity and accuracy of these approximate distributions. These
approximate probability distributions can be used to valuate two-asset options,
e.g. spread options and basket
options, where the CEV variables represent the forward prices of the underlying
assets. Moreover, we believe that this new approach can be extended to study
the algebraic sum of *N* CEV variables
with potential applications in pricing multi-asset options.

Cite this paper

Lo, C. (2013) The Sum and Difference of Two Constant Elasticity of Variance Stochastic Variables.*Applied Mathematics*, **4**, 1503-1511. doi: 10.4236/am.2013.411203.

Lo, C. (2013) The Sum and Difference of Two Constant Elasticity of Variance Stochastic Variables.

References

[1] C. F. Lo, “The Sum and Difference of Two Lognormal Random Variables,” Journal of Applied Mathematics, Vol. 2012, 2012, Article ID: 838397.

http://dx.doi.org/10.1155/2012/838397

[2] L. Fenton, “The Sum of Lognormal Probability Distributions in Scatter Transmission Systems,” IRE Transactions on Communications Systems, Vol. 8, No. 1, 1960, pp. 57-67.

http://dx.doi.org/10.1109/TCOM.1960.1097606

[3] J. I. Naus, “The Distribution of the Logarithm of the Sum of Two Lognormal Variates,” Journal of the American Statistical Association, Vol. 64, No. 326, 1969, pp. 655-659.

http://dx.doi.org/10.1080/01621459.1969.10501004

[4] M. A. Hamdan, “The Logarithm of the Sum of Two Correlated Lognormal Variates,” Journal of the American Statistical Association, Vol. 66, No. 333, 1971, pp. 105-106.

http://dx.doi.org/10.1080/01621459.1971.10482229

[5] C. L. Ho, “Calculating the Mean and Variance of Power Sums with Two Lognormal Components,” IEEE Transactions on Vehicular Technology, Vol. 44, No. 4, 1995, pp. 756-762.

http://dx.doi.org/10.1109/25.467959

[6] M. A. Milevsky and S. E. Posner, “Asian Options, the Sum of Lognormals, and the Reciprocal Gamma Distribution,” Journal of Financial and Quantitative Analysis, Vol. 33, No. 3, 1998, pp. 409-422.

http://dx.doi.org/10.2307/2331102

[7] J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas and D. Vyncke, “The Concept of Comonotonicity in Actuarial Science and Finance: Applications,” Insurance: Mathematics and Economics, Vol. 31, No. 2, 2002, pp. 133-161.

http://dx.doi.org/10.1016/S0167-6687(02)00135-X

[8] R. Carmona and V. Durrleman, “Pricing and Hedging Spread Options,” SIAM Review, Vol. 45, No. 4, 2003, pp. 627-685. http://dx.doi.org/10.1137/S0036144503424798

[9] J. H. Graham, K. Shimizu, J. M. Emlen, D. C. Freeman and J. Merkel, “Growth Models and the Expected Distribution of Fluctuating Asymmetry,” Biological Journal of the Linnean Society, Vol. 80, No. 1, 2003, pp. 57-65.

http://dx.doi.org/10.1046/j.1095-8312.2003.00220.x

[10] M. Romeo, V. Da Costa and F. Bardon, “Broad Distribution Effects in Sums of Lognormal Random Variables,” European Physical Journal B, Vol. 32, No. 4, 2003, pp. 513-525.

http://dx.doi.org/10.1140/epjb/e2003-00131-6

[11] D. Dufresne, “The Log-Normal Approximation in Financial and Other Computations,” Advances in Applied Probability, Vol. 36, No. 3, 2004, pp. 747-773.

http://dx.doi.org/10.1239/aap/1093962232

[12] S. Vanduffel, T. Hoedemakers and J. Dhaene, “Comparing Approximations for Risk Measures of Sums of NonIndependent Lognormal Random Variables,” North American Actuarial Journal, Vol. 9, No. 4, 2005, pp. 71-82.

[13] J. Wu, N. B. Mehta and J. Zhang, “Flexible Lognormal Sum Approximation Method,” Proceedings of IEEE Global Telecommunications Conference (GLOBECOM’05), Vol. 6, 2005, pp. 3413-3417.

[14] A. Kukush and M. Pupashenko, “Bounds for a Sum of Random Variables under a Mixture of Normals,” Theory of Stochastic Processes, Vol. 13, No. 29, 2007, pp. 82-97.

[15] X. Gao, H. Xu and D. Ye, “Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables,” International Journal of Mathematics and Mathematical Sciences, Vol. 2009, 2009, Article ID: 630857.

[16] H. F. Trotter, “On the Product of Semi-Groups of Operators,” Proceedings of the American Mathematical Society, Vol. 10, No. 4, 1959, pp. 545-551.

http://dx.doi.org/10.1090/S0002-9939-1959-0108732-6

[17] N. C. Beaulieu and F. Rajwani, “Highly Accurate Simple Closed-Form Approximations to Lognormal Sum Distributions and Densities,” IEEE Communications Letters, Vol. 8, No. 12, 2004, pp. 709-711.

http://dx.doi.org/10.1109/LCOMM.2004.837657

[18] C. L. J. Lam and T. Le-Ngoc, “Estimation of Typical Sum of Lognormal Random Variables Using Log-Shifted Gamma Approximation,” IEEE Communications Letters, Vol. 10, No. 4, 2006, pp. 234-235.

http://dx.doi.org/10.1109/LCOMM.2006.1613731

[19] L. Zhao and J. Ding, “Least Squares Approximations to Lognormal Sum Distributions,” IEEE Transactions on Vehicular Technology, Vol. 56, No. 2, 2007, pp. 991-997.

http://dx.doi.org/10.1109/TVT.2007.891467

[20] N. B. Mehta, J. Wu, A. F. Molisch and J. Zhang, “Approximating a Sum of Random Variables with a Lognormal,” IEEE Transactions on Wireless Communications, Vol. 6, No. 7, 2007, pp. 2690-2699.

http://dx.doi.org/10.1109/TWC.2007.051000

[21] S. Borovkova, F. J. Permana and H. V. D. Weide, “A Closed Form Approach to the Valuation and Hedging of Basket and Spread Options,” The Journal of Derivatives, Vol. 14, No. 4, 2007, pp. 8-24.

http://dx.doi.org/10.3905/jod.2007.686420

[22] D. Dufresne, “Sums of Lognormals,” Actuarial Research Conference Proceedings, Regina, 14-16 August 2008.

[23] Q. T. Zhang and S. H. Song, “A Systematic Procedure for Accurately Approximating Lognormal-Sum Distributions,” IEEE Transactions on Vehicular Technology, Vol. 57, No. 1, 2008, pp. 663-666.

http://dx.doi.org/10.1109/TVT.2007.905611

[24] T. R. Hurd and Z. Zhou, “A Fourier Transform Method for Spread Option Pricing,” SIAM Journal of Financial Mathematics, Vol. 1, No. 1, 2010, pp. 142-157.

http://dx.doi.org/10.1137/090750421

[25] X. Li, V. D. Chakravarthy and Z. Wu, “A Low-Complexity Approximation to Lognormal Sum Distributions via Transformed Log Skew Normal Distribution,” IEEE Transactions on Vehicular Technology, Vol. 60, No. 8, 2011, pp. 4040-4045.

http://dx.doi.org/10.1109/TVT.2011.2163652

[26] N. C. Beaulieu, “An Extended Limit Theorem for Correlated Lognormal Sums,” IEEE Transactions on Communications, Vol. 60, No. 1, 2012, pp. 23-26.

http://dx.doi.org/10.1109/TCOMM.2011.091911.110054

[27] J. J. Chang, S. N. Chen and T. P. Wu, “A Note to Enhance the BPW Model for the Pricing of Basket and Spread Options,” The Journal of Derivatives, Vol. 19, No. 3, 2012, pp. 77-82.

http://dx.doi.org/10.3905/jod.2012.19.3.077

[28] J. C. Cox and S. A. Ross, “The Valuation of Options for Alternative Stochastic Processes,” Journal of Financial Economics, Vol. 3, No. 1-2, 1976, pp. 145-166.

http://dx.doi.org/10.1016/0304-405X(76)90023-4

[29] J. C. Cox, “The Constant Elasticity of Variance Option Pricing Model,” Journal of Portfolio Management, Vol. 23, 1996, pp. 15-17.

[30] C. F. Lo, P. H. Yuen and C. H. Hui, “Constant Elasticity of Variance Option Pricing Model with Time-Dependent Parameters,” International Journal of Theoretical and Applied Finance, Vol. 3, No. 4, 2000, pp. 661-674.

http://dx.doi.org/10.1142/S0219024900000814

[31] D. Davydov and V. Linetsky, “The Valuation and Hedging of Barrier and Lookback Option under the CEV Process,” Management Science, Vol. 47, 2001, pp. 949-965.

http://dx.doi.org/10.1287/mnsc.47.7.949.9804

[32] J. Detemple and W. D. Tian, “The Valuation of American Options for a Class of Diffusion Processes,” Management Science, Vol. 48, No. 7, 2002, pp. 917-937.

http://dx.doi.org/10.1287/mnsc.48.7.917.2815

[33] C. Jones, “The Dynamics of the Stochastic Volatility: Evidence from Underlying and Options Markets,” Journal of Econometrics, Vol. 116, No. 1-2, 2003, pp. 181-224. http://dx.doi.org/10.1016/S0304-4076(03)00107-6

[34] M. Widdicks, P. W. Duck, A. D. Andricopoulos and D. P. Newton, “The Black-Scholes Equation Revisited: Asymptotic Expansions and Singular Perturbations,” Mathematical Finance, Vol. 15, No. 2, 2005, pp. 373-391.

http://dx.doi.org/10.1111/j.0960-1627.2005.00224.x

[35] J. Xiao, H. Zhai and C. Qin, “The Constant Elasticity of Variance (CEV) Model and the Legendre Transform— Dual Solution for Annuity Contracts,” Insurance: Mathematics and Economics, Vol. 40, 2007, pp. 302-310.

http://dx.doi.org/10.1016/j.insmatheco.2006.04.007

[36] Y. L. Hsu, T. I. Lin and C. F. Lee, “Constant Elasticity of Variance (CEV) Option Pricing Model: Integration and Detailed Derivation,” Mathematics and Computers in Simulation, Vol. 79, No. 1, 2008, pp. 60-71.

http://dx.doi.org/10.1016/j.matcom.2007.09.012

[37] R. R. Chen, C. F. Lee and H. H. Lee, “Empirical Performance of the Constant Elasticity Variance Option Pricing Model,” Review of Pacific Basin Financial Markets and Policies, Vol. 12, No. 2, 2009, pp. 1-41.

http://dx.doi.org/10.1142/S0219091509001605

[38] J. Gao, “Optimal Portfolios for DC Pension Plans under a CEV Model,” Insurance: Mathematics and Economics, Vol. 44, No. 3, 2009, pp. 479-490.

http://dx.doi.org/10.1016/j.insmatheco.2009.01.005

[39] M. Gu, Y. Yang, S. Li and J. Zhang, “Constant Elasticity of Variance Model for Proportional Reinsurance and Investment Strategies,” Insurance: Mathematics and Economics, Vol. 46, No. 3, 2010, pp. 9-18.

[40] X. Lin and Y. Li, “Optimal Reinsurance and Investment for a Jump Diffusion Risk Process under the CEV Model,” North American Actuarial Journal, Vol. 15, No. 3, 2011, pp. 417-431.

http://dx.doi.org/10.1080/10920277.2011.10597628

[41] H. Zhao and X. Rong, “Portfolio Selection Problem with Multiple Risky Assets under the Constant Elasticity of Variance Model,” Insurance: Mathematics and Economics, Vol. 50, No. 1, 2012, pp. 179-190.

http://dx.doi.org/10.1016/j.insmatheco.2011.10.013

[42] J. E. Ingersoll, “Theory of Financial Decision Making,” Rowman & Littlefield Publishers, Lanham, 1987.

[43] D. R. Cox and H. D. Miller, “Theory of Stochastic Processes,” Chapman & Hall/CRC, London, 2001.

[44] J. Baz and G. Chacko, “Financial Derivatives: Pricing, Applications and Mathematics,” Cambridge University Press, Cambridge, 2009.

[45] G. W. Gardiner, “Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences,” Springer, Berlin, 2004.

[46] H. F. Trotter, “Approximation of Semi-Groups of Operators,” Pacific Journal of Mathematic, Vol. 8, No. 4, 1958, pp. 887-919. http://dx.doi.org/10.2140/pjm.1958.8.887

[47] M. Suzuki, “Decomposition Formulas of Exponential Operators and Lie Exponentials with Some Applications to Quantum Mechanics and Statistical Physics,” Journal of Mathematical Physics, Vol. 26, No. 4, 1985, pp. 601-612. http://dx.doi.org/10.1063/1.526596

[48] A. N. Drozdov and J. J. Brey, “Operator Expansions in Stochastic Dynamics,” Physical Review E, Vol. 57, No. 2, 1998, pp. 1284-1289.

http://dx.doi.org/10.1103/PhysRevE.57.1284

[49] N. Hatano and M. Suzuki, “Finding Exponential Product Formulas of Higher Orders,” Lecture Notes in Physics, Vol. 679, 2005, pp. 37-68.

[1] C. F. Lo, “The Sum and Difference of Two Lognormal Random Variables,” Journal of Applied Mathematics, Vol. 2012, 2012, Article ID: 838397.

http://dx.doi.org/10.1155/2012/838397

[2] L. Fenton, “The Sum of Lognormal Probability Distributions in Scatter Transmission Systems,” IRE Transactions on Communications Systems, Vol. 8, No. 1, 1960, pp. 57-67.

http://dx.doi.org/10.1109/TCOM.1960.1097606

[3] J. I. Naus, “The Distribution of the Logarithm of the Sum of Two Lognormal Variates,” Journal of the American Statistical Association, Vol. 64, No. 326, 1969, pp. 655-659.

http://dx.doi.org/10.1080/01621459.1969.10501004

[4] M. A. Hamdan, “The Logarithm of the Sum of Two Correlated Lognormal Variates,” Journal of the American Statistical Association, Vol. 66, No. 333, 1971, pp. 105-106.

http://dx.doi.org/10.1080/01621459.1971.10482229

[5] C. L. Ho, “Calculating the Mean and Variance of Power Sums with Two Lognormal Components,” IEEE Transactions on Vehicular Technology, Vol. 44, No. 4, 1995, pp. 756-762.

http://dx.doi.org/10.1109/25.467959

[6] M. A. Milevsky and S. E. Posner, “Asian Options, the Sum of Lognormals, and the Reciprocal Gamma Distribution,” Journal of Financial and Quantitative Analysis, Vol. 33, No. 3, 1998, pp. 409-422.

http://dx.doi.org/10.2307/2331102

[7] J. Dhaene, M. Denuit, M. J. Goovaerts, R. Kaas and D. Vyncke, “The Concept of Comonotonicity in Actuarial Science and Finance: Applications,” Insurance: Mathematics and Economics, Vol. 31, No. 2, 2002, pp. 133-161.

http://dx.doi.org/10.1016/S0167-6687(02)00135-X

[8] R. Carmona and V. Durrleman, “Pricing and Hedging Spread Options,” SIAM Review, Vol. 45, No. 4, 2003, pp. 627-685. http://dx.doi.org/10.1137/S0036144503424798

[9] J. H. Graham, K. Shimizu, J. M. Emlen, D. C. Freeman and J. Merkel, “Growth Models and the Expected Distribution of Fluctuating Asymmetry,” Biological Journal of the Linnean Society, Vol. 80, No. 1, 2003, pp. 57-65.

http://dx.doi.org/10.1046/j.1095-8312.2003.00220.x

[10] M. Romeo, V. Da Costa and F. Bardon, “Broad Distribution Effects in Sums of Lognormal Random Variables,” European Physical Journal B, Vol. 32, No. 4, 2003, pp. 513-525.

http://dx.doi.org/10.1140/epjb/e2003-00131-6

[11] D. Dufresne, “The Log-Normal Approximation in Financial and Other Computations,” Advances in Applied Probability, Vol. 36, No. 3, 2004, pp. 747-773.

http://dx.doi.org/10.1239/aap/1093962232

[12] S. Vanduffel, T. Hoedemakers and J. Dhaene, “Comparing Approximations for Risk Measures of Sums of NonIndependent Lognormal Random Variables,” North American Actuarial Journal, Vol. 9, No. 4, 2005, pp. 71-82.

[13] J. Wu, N. B. Mehta and J. Zhang, “Flexible Lognormal Sum Approximation Method,” Proceedings of IEEE Global Telecommunications Conference (GLOBECOM’05), Vol. 6, 2005, pp. 3413-3417.

[14] A. Kukush and M. Pupashenko, “Bounds for a Sum of Random Variables under a Mixture of Normals,” Theory of Stochastic Processes, Vol. 13, No. 29, 2007, pp. 82-97.

[15] X. Gao, H. Xu and D. Ye, “Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables,” International Journal of Mathematics and Mathematical Sciences, Vol. 2009, 2009, Article ID: 630857.

[16] H. F. Trotter, “On the Product of Semi-Groups of Operators,” Proceedings of the American Mathematical Society, Vol. 10, No. 4, 1959, pp. 545-551.

http://dx.doi.org/10.1090/S0002-9939-1959-0108732-6

[17] N. C. Beaulieu and F. Rajwani, “Highly Accurate Simple Closed-Form Approximations to Lognormal Sum Distributions and Densities,” IEEE Communications Letters, Vol. 8, No. 12, 2004, pp. 709-711.

http://dx.doi.org/10.1109/LCOMM.2004.837657

[18] C. L. J. Lam and T. Le-Ngoc, “Estimation of Typical Sum of Lognormal Random Variables Using Log-Shifted Gamma Approximation,” IEEE Communications Letters, Vol. 10, No. 4, 2006, pp. 234-235.

http://dx.doi.org/10.1109/LCOMM.2006.1613731

[19] L. Zhao and J. Ding, “Least Squares Approximations to Lognormal Sum Distributions,” IEEE Transactions on Vehicular Technology, Vol. 56, No. 2, 2007, pp. 991-997.

http://dx.doi.org/10.1109/TVT.2007.891467

[20] N. B. Mehta, J. Wu, A. F. Molisch and J. Zhang, “Approximating a Sum of Random Variables with a Lognormal,” IEEE Transactions on Wireless Communications, Vol. 6, No. 7, 2007, pp. 2690-2699.

http://dx.doi.org/10.1109/TWC.2007.051000

[21] S. Borovkova, F. J. Permana and H. V. D. Weide, “A Closed Form Approach to the Valuation and Hedging of Basket and Spread Options,” The Journal of Derivatives, Vol. 14, No. 4, 2007, pp. 8-24.

http://dx.doi.org/10.3905/jod.2007.686420

[22] D. Dufresne, “Sums of Lognormals,” Actuarial Research Conference Proceedings, Regina, 14-16 August 2008.

[23] Q. T. Zhang and S. H. Song, “A Systematic Procedure for Accurately Approximating Lognormal-Sum Distributions,” IEEE Transactions on Vehicular Technology, Vol. 57, No. 1, 2008, pp. 663-666.

http://dx.doi.org/10.1109/TVT.2007.905611

[24] T. R. Hurd and Z. Zhou, “A Fourier Transform Method for Spread Option Pricing,” SIAM Journal of Financial Mathematics, Vol. 1, No. 1, 2010, pp. 142-157.

http://dx.doi.org/10.1137/090750421

[25] X. Li, V. D. Chakravarthy and Z. Wu, “A Low-Complexity Approximation to Lognormal Sum Distributions via Transformed Log Skew Normal Distribution,” IEEE Transactions on Vehicular Technology, Vol. 60, No. 8, 2011, pp. 4040-4045.

http://dx.doi.org/10.1109/TVT.2011.2163652

[26] N. C. Beaulieu, “An Extended Limit Theorem for Correlated Lognormal Sums,” IEEE Transactions on Communications, Vol. 60, No. 1, 2012, pp. 23-26.

http://dx.doi.org/10.1109/TCOMM.2011.091911.110054

[27] J. J. Chang, S. N. Chen and T. P. Wu, “A Note to Enhance the BPW Model for the Pricing of Basket and Spread Options,” The Journal of Derivatives, Vol. 19, No. 3, 2012, pp. 77-82.

http://dx.doi.org/10.3905/jod.2012.19.3.077

[28] J. C. Cox and S. A. Ross, “The Valuation of Options for Alternative Stochastic Processes,” Journal of Financial Economics, Vol. 3, No. 1-2, 1976, pp. 145-166.

http://dx.doi.org/10.1016/0304-405X(76)90023-4

[29] J. C. Cox, “The Constant Elasticity of Variance Option Pricing Model,” Journal of Portfolio Management, Vol. 23, 1996, pp. 15-17.

[30] C. F. Lo, P. H. Yuen and C. H. Hui, “Constant Elasticity of Variance Option Pricing Model with Time-Dependent Parameters,” International Journal of Theoretical and Applied Finance, Vol. 3, No. 4, 2000, pp. 661-674.

http://dx.doi.org/10.1142/S0219024900000814

[31] D. Davydov and V. Linetsky, “The Valuation and Hedging of Barrier and Lookback Option under the CEV Process,” Management Science, Vol. 47, 2001, pp. 949-965.

http://dx.doi.org/10.1287/mnsc.47.7.949.9804

[32] J. Detemple and W. D. Tian, “The Valuation of American Options for a Class of Diffusion Processes,” Management Science, Vol. 48, No. 7, 2002, pp. 917-937.

http://dx.doi.org/10.1287/mnsc.48.7.917.2815

[33] C. Jones, “The Dynamics of the Stochastic Volatility: Evidence from Underlying and Options Markets,” Journal of Econometrics, Vol. 116, No. 1-2, 2003, pp. 181-224. http://dx.doi.org/10.1016/S0304-4076(03)00107-6

[34] M. Widdicks, P. W. Duck, A. D. Andricopoulos and D. P. Newton, “The Black-Scholes Equation Revisited: Asymptotic Expansions and Singular Perturbations,” Mathematical Finance, Vol. 15, No. 2, 2005, pp. 373-391.

http://dx.doi.org/10.1111/j.0960-1627.2005.00224.x

[35] J. Xiao, H. Zhai and C. Qin, “The Constant Elasticity of Variance (CEV) Model and the Legendre Transform— Dual Solution for Annuity Contracts,” Insurance: Mathematics and Economics, Vol. 40, 2007, pp. 302-310.

http://dx.doi.org/10.1016/j.insmatheco.2006.04.007

[36] Y. L. Hsu, T. I. Lin and C. F. Lee, “Constant Elasticity of Variance (CEV) Option Pricing Model: Integration and Detailed Derivation,” Mathematics and Computers in Simulation, Vol. 79, No. 1, 2008, pp. 60-71.

http://dx.doi.org/10.1016/j.matcom.2007.09.012

[37] R. R. Chen, C. F. Lee and H. H. Lee, “Empirical Performance of the Constant Elasticity Variance Option Pricing Model,” Review of Pacific Basin Financial Markets and Policies, Vol. 12, No. 2, 2009, pp. 1-41.

http://dx.doi.org/10.1142/S0219091509001605

[38] J. Gao, “Optimal Portfolios for DC Pension Plans under a CEV Model,” Insurance: Mathematics and Economics, Vol. 44, No. 3, 2009, pp. 479-490.

http://dx.doi.org/10.1016/j.insmatheco.2009.01.005

[39] M. Gu, Y. Yang, S. Li and J. Zhang, “Constant Elasticity of Variance Model for Proportional Reinsurance and Investment Strategies,” Insurance: Mathematics and Economics, Vol. 46, No. 3, 2010, pp. 9-18.

[40] X. Lin and Y. Li, “Optimal Reinsurance and Investment for a Jump Diffusion Risk Process under the CEV Model,” North American Actuarial Journal, Vol. 15, No. 3, 2011, pp. 417-431.

http://dx.doi.org/10.1080/10920277.2011.10597628

[41] H. Zhao and X. Rong, “Portfolio Selection Problem with Multiple Risky Assets under the Constant Elasticity of Variance Model,” Insurance: Mathematics and Economics, Vol. 50, No. 1, 2012, pp. 179-190.

http://dx.doi.org/10.1016/j.insmatheco.2011.10.013

[42] J. E. Ingersoll, “Theory of Financial Decision Making,” Rowman & Littlefield Publishers, Lanham, 1987.

[43] D. R. Cox and H. D. Miller, “Theory of Stochastic Processes,” Chapman & Hall/CRC, London, 2001.

[44] J. Baz and G. Chacko, “Financial Derivatives: Pricing, Applications and Mathematics,” Cambridge University Press, Cambridge, 2009.

[45] G. W. Gardiner, “Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences,” Springer, Berlin, 2004.

[46] H. F. Trotter, “Approximation of Semi-Groups of Operators,” Pacific Journal of Mathematic, Vol. 8, No. 4, 1958, pp. 887-919. http://dx.doi.org/10.2140/pjm.1958.8.887

[47] M. Suzuki, “Decomposition Formulas of Exponential Operators and Lie Exponentials with Some Applications to Quantum Mechanics and Statistical Physics,” Journal of Mathematical Physics, Vol. 26, No. 4, 1985, pp. 601-612. http://dx.doi.org/10.1063/1.526596

[48] A. N. Drozdov and J. J. Brey, “Operator Expansions in Stochastic Dynamics,” Physical Review E, Vol. 57, No. 2, 1998, pp. 1284-1289.

http://dx.doi.org/10.1103/PhysRevE.57.1284

[49] N. Hatano and M. Suzuki, “Finding Exponential Product Formulas of Higher Orders,” Lecture Notes in Physics, Vol. 679, 2005, pp. 37-68.